I tried several of the methods identified by others, but they did not work for me. I, too, had used Office 2003 satisfactorily on Windows 10 Home. I bought a new computer (a 17" HP with intel i5, 1 T SSD and 16 G of ram) running Windows 11 home. I found that Word and Excel both had problems with hanging on open and save as.
I found a work around was to locate a saved file from previous work, copy the file and rename it, delete the contents, and load the new content for the file either by typing it in or copying it from a text editor. That way save would work as long as I didn't press save as. I resigned myself to doing that until I could find a solution.
Shortly after that I noticed that OneDrive was set up to receive my files from <documents>. Then I discovered that all of my files were being pointed to the cloud in Windows 11. I went onto the internet to find out how to stop that - I am one of those people who does not want to put my stuff on the cloud. I found out how to do it and stopped OneDrive from being the repository then moved everything that was saved there off the OneDrive, back to the c: drive.
I went back to work and lo and behold Word 2003 and Excel 2003 are now able to operate correctly in both open and save as. It appears that somehow, the presence of OneDrive was impeding these operations. And now they are working as expected and as they had on Windows 10 previously.
I am sure that someone else can explain this and I for one am hard pressed to imagine that MS is unaware of it.